Sharp upper bounds for the capacity in the hyperbolic and Euclidean spaces - Li Haizhong; Li Ruixuan; Xiong Changwei (Advances in Nonlinear Analysis)
We derive various sharp upper bounds for the pp-capacity of a smooth compact set KK in the hyperbolic space Hn{{\mathbb{H}}}^{n} and the Euclidean space Rn{{\mathbb{R}}}^{n}. First, by using the inverse mean curvature flow, for the mean convex and star-shaped set KK in Hn{{\mathbb{H}}}^{n}, we obtain sharp upper bounds for the pp-capacity Capp(K){{\rm{Cap}}}_{p}\left(K) in three cases: (1) n≥2n\ge 2 and p=2p=2, (2) n=2n=2 and p≥3p\ge 3, and (3) n=3n=3 and 1<p≤31\lt p\le 3; using the unit-speed normal flow, we prove a sharp upper bound for Capp(K){{\rm{Cap}}}_{p}\left(K) of a convex set KK in Hn{{\mathbb{H}}}^{n} for n≥2n\ge 2 and p>1p\gt 1. Second, for the compact set KK in R3{{\mathbb{R}}}^{3}, using the weak inverse mean curvature flow, we obtain a sharp upper bound for the pp-capacity (1<p<31\lt p\lt 3) of the set KK with connected boundary; by using the inverse anisotropic mean curvature flow, we deduce a sharp upper bound for the anisotropic pp-capacity (1<p<31\lt p\lt 3) of an FF-mean convex and star-shaped set KK in R3{{\mathbb{R}}}^{3}.